Group #24907322 — subgroups

Subgroups — #24907322

Order 10624.

260 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₈₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581702	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	#19127547	C₂	1	1	yes	maximal, characteristic
5	#19127455	C₂	1	1	yes	maximal, characteristic
6	#19127455	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
12	C₄⋊(C₈₃⋊C₈)	C₄	1	4	yes	complemented
13	#14581702	—	2	2	no	—
14	C₈×Dic₈₃	—	2	2	no	—
15	C₈₃⋊#95	C₄	1	2	yes	—
16	C₈₃⋊#95	C₄	1	2	yes	—
17	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₈₃	C₄	1	4	yes	complemented
21	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
22	C₂²×Dic₈₃	C₂×C₄	1	2	yes	—
23	C₂×C₈₃⋊C₈	—	2	1	no	—
24	C₂×C₈₃⋊C₈	—	2	1	no	—
25	C₂×C₈₃⋊C₈	—	2	1	no	—
26	C₂×C₈₃⋊C₈	—	2	1	no	—
27	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
29	C₄×Dic₈₃	—	2	1	no	—
30	C₄×Dic₈₃	—	2	1	no	—
31	C₄×Dic₈₃	D₄	1	2	yes	—
32	C₄×Dic₈₃	Q₈	1	2	yes	—
33	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₈₃	—	2	1	no	—
37	C₂×C₈×C₈₃	—	2	1	no	—
38	C₂×C₈×C₈₃	—	2	1	no	—
39	C₂×C₈×C₈₃	—	2	1	no	—
40	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₈₃	C₂×C₄	1	2	yes	—
42	C₄²×C₈₃	C₂×C₄	1	2	yes	—
43	C₄²×C₈₃	C₂×C₄	1	2	yes	—
44	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	—	2	2	no	—
46	C₂×C₄×C₈₃	—	2	2	no	—
47	C₂×C₄×C₈₃	C₄²	1	2	yes	—
48	C₂×C₄×C₈₃	C₄²	1	2	yes	—
49	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
55	C₂×Dic₈₃	—	2	2	no	—
56	C₂×Dic₈₃	—	2	2	no	—
57	C₂×Dic₈₃	—	2	2	no	—
58	C₂×Dic₈₃	—	2	2	no	—
59	C₂×Dic₈₃	—	2	2	no	—
60	C₂×Dic₈₃	C₄⋊C₄	1	4	yes	—
61	C₈×C₈₃	—	2	2	no	—
62	C₈×C₈₃	—	2	2	no	—
63	C₈₃⋊C₈	—	2	2	no	—
64	C₈₃⋊C₈	—	2	2	no	—
65	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₈₃	#117	1	1	yes	characteristic
68	C₂²×C₈₃	#131	1	2	yes	—
69	C₂²×C₈₃	#131	1	2	yes	—
70	C₄×C₈₃	—	2	2	no	—
71	C₄×C₈₃	—	2	2	no	—
72	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
73	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
74	C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
75	C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
76	Dic₈₃	—	2	4	no	—
77	Dic₈₃	—	4	2	no	—
78	C₂×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₈₃	#405	1	1	yes	characteristic
80	C₂×C₈₃	#405	1	1	yes	characteristic
81	C₂×C₈₃	#424	1	2	yes	—
82	C₂×C₈₃	#433	1	2	yes	—
83	#1768	—	83	1	no	maximal, Sylow
84	C₈₃	#1768	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊C₈	—	83	1	no	—
86	C₂×C₄⋊C₈	D₈₃	1	1	yes	characteristic
87	C₂×#97	—	83	1	no	—
88	C₂×C₄×C₈	—	83	1	no	—
89	#376	—	83	1	no	—
90	#336	—	83	1	no	—
91	#336	—	83	1	no	—
92	C₂²×C₈	—	83	1	no	—
93	C₂²×C₈	—	83	1	no	—
94	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₈₃	1	1	yes	characteristic
96	C₂×C₄²	—	83	1	no	—
97	C₂×C₄²	—	83	1	no	—
98	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
99	C₄⋊C₈	—	83	4	no	—
100	C₄⋊C₈	Dic₈₃	1	4	yes	complemented
101	#97	—	166	2	no	—
102	C₄×C₈	—	166	2	no	—
103	#95	—	83	2	no	—
104	#95	—	83	2	no	—
105	C₂²×C₄	—	83	2	no	—
106	C₂²×C₄	—	83	2	no	—
107	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	83	2	no	—
115	C₂×C₈	—	83	2	no	—
116	C₂×C₈	—	166	1	no	—
117	C₂×C₈	—	166	1	no	—
118	C₂×C₈	—	166	1	no	—
119	C₂×C₈	—	166	1	no	—
120	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
121	C₂×C₈	C₂×Dic₈₃	1	2	yes	—
122	C₄²	—	83	2	no	—
123	C₄²	—	83	2	no	—
124	C₄²	—	166	1	no	—
125	C₄²	—	166	1	no	—
126	C₄²	C₄×D₈₃	1	2	yes	—
127	C₄²	C₂×Dic₈₃	1	2	yes	—
128	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	83	4	no	—
132	C₂×C₄	—	166	2	no	—
133	C₂×C₄	—	166	2	no	—
134	C₂×C₄	—	166	2	no	—
135	C₂×C₄	—	166	2	no	—
136	C₂×C₄	—	166	2	no	—
137	C₂×C₄	C₄×Dic₈₃	1	2	yes	—
138	C₂×C₄	C₄×Dic₈₃	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
140	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
141	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₈₃	1	1	yes	characteristic
143	C₂×C₄	#13050589	1	1	yes	characteristic
144	C₂×C₄	#13050591	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	166	2	no	—
148	C₈	—	166	2	no	—
149	C₂²	C₂×C₄×Dic₈₃	1	1	yes	characteristic
150	C₂²	C₄⋊C₄×D₈₃	1	1	yes	characteristic
151	C₂²	#14581750	1	1	yes	characteristic
152	C₂²	#14581798	1	2	yes	—
153	C₂²	#14581800	1	2	yes	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	332	2	no	—
157	C₄	—	166	4	no	—
158	C₄	#14581754	1	1	yes	characteristic
159	C₄	D₄×Dic₈₃	1	1	yes	characteristic
160	C₄	#14581757	1	1	yes	characteristic
161	C₄	Q₈×Dic₈₃	1	1	yes	characteristic
162	C₂	C₄⋊C₄×Dic₈₃	1	1	yes	characteristic
163	C₂	#19127656	1	1	yes	characteristic
164	C₂	#19127659	1	1	yes	characteristic
165	C₂	#19127726	1	2	yes	—
166	C₂	#19127761	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₈₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14581702

C₂

yes

maximal, characteristic

C₂×C₈×Dic₈₃

C₂

yes

maximal, characteristic

#19127547

C₂

yes

maximal, characteristic

#19127455

C₂

yes

maximal, characteristic

#19127455

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₈₃

C₂

yes

maximal, characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

C₂×C₄×Dic₈₃

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₄

yes

complemented

#14581702

—

C₈×Dic₈₃

—

C₈₃⋊#95

C₄

yes

—

C₈₃⋊#95

C₄

yes

—

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂²×C₈×C₈₃

C₂²

yes

characteristic

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

C₄

yes

complemented

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂²×Dic₈₃

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

D₄

yes

—

C₄×Dic₈₃

Q₈

yes

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

—

C₄²×C₈₃

C₂×C₄

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

—

C₂×C₄×C₈₃

C₄²

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

C₄⋊C₄

yes

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂×C₄²

yes

characteristic

C₂²×C₈₃

C₂×C₄⋊C₄

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₂²×C₈₃

#131

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

C₄×D₄

yes

characteristic

C₄×C₈₃

C₄×Q₈

yes

characteristic

C₄×C₈₃

C₄×Q₈

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

C₄×C₄⋊C₄

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

#433

yes

—

#1768

—

maximal, Sylow

C₈₃

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₈₃

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂²×C₈

C₂×D₈₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

Dic₈₃

yes

complemented

101

#97

—

166

—

102

C₄×C₈

—

166

—

103

#95

—

104

#95

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₈₃

yes

characteristic

108

C₂²×C₄

C₄×D₈₃

yes

characteristic

109

C₂²×C₄

C₂²×D₈₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

166

—

117

C₂×C₈

—

166

—

118

C₂×C₈

—

166

—

119

C₂×C₈

—

166

—

120

C₂×C₈

C₂×Dic₈₃

yes

—

121

C₂×C₈

C₂×Dic₈₃

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

166

—

125

C₄²

—

166

—

126

C₄²

C₄×D₈₃

yes

—

127

C₄²

C₂×Dic₈₃

yes

—

128

C₂³

C₂×C₄×D₈₃

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

166

—

133

C₂×C₄

—

166

—

134

C₂×C₄

—

166

—

135

C₂×C₄

—

166

—

136

C₂×C₄

—

166

—

137

C₂×C₄

C₄×Dic₈₃

yes

—

138

C₂×C₄

C₄×Dic₈₃

yes

—

139

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

140

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

141

C₂×C₄

D₄×D₈₃

yes

characteristic

142

C₂×C₄

Q₈×D₈₃

yes

characteristic

143

C₂×C₄

#13050589

yes

characteristic

144

C₂×C₄

#13050591

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

166

—

148

C₈

—

166

—

149

C₂²

C₂×C₄×Dic₈₃

yes

characteristic

150

C₂²

C₄⋊C₄×D₈₃

yes

characteristic

151

C₂²

#14581750

yes

characteristic

152

C₂²

#14581798

yes

—

153

C₂²

#14581800

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

332

—

157

C₄

—

166

—

158

C₄

#14581754

yes

characteristic

159

C₄

D₄×Dic₈₃

yes

characteristic

160

C₄

#14581757

yes

characteristic

161

C₄

Q₈×Dic₈₃

yes

characteristic

162

C₂

C₄⋊C₄×Dic₈₃

yes

characteristic

163

C₂

#19127656

yes

characteristic

164

C₂

#19127659

yes

characteristic

165

C₂

#19127726

yes

—

166

C₂

#19127761

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₃
commutator	C₂²×C₈₃	C₂×C₄²
Fitting	C₂×C₄⋊C₈×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₈₃
hypocenter	C₈₃	#1768
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₃

commutator

C₂²×C₈₃

C₂×C₄²

Fitting

C₂×C₄⋊C₈×C₈₃

C₂

socle, prime radical

C₂³×C₈₃

C₂²×C₄

cosocle

C₂²×C₄×C₈₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₈₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
12. C₄⋊(C₈₃⋊C₈)	C₄	semidirect
20. C₄⋊C₈×C₈₃	C₄	semidirect
84. C₈₃	#1768	semidirect
100. C₄⋊C₈	Dic₈₃	semidirect

H ≅ G/N

type

12. C₄⋊(C₈₃⋊C₈)

C₄

semidirect

20. C₄⋊C₈×C₈₃

C₄

semidirect

84. C₈₃

#1768

semidirect

100. C₄⋊C₈

Dic₈₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊(C₈₃⋊C₈)	2
C₂×#14581702	2
C₂×C₈×Dic₈₃	2
#19127547	2
#19127455	2
C₂×C₄⋊C₈×C₈₃	2
#1768	83

subgroup

index

C₂×C₄⋊(C₈₃⋊C₈)

C₂×C₈×Dic₈₃

C₂×C₄⋊C₈×C₈₃

Subgroups — #24907322

Conjugacy classes of subgroups

Complements

Maximal subgroups